Baseball ProGUESTus: The Physics of a Truly Amazing Throw

Alan Nathan is Professor Emeritus of Physics at the University of Illinois at Urbana-Champaign. After a long career doing things like measuring the electric and magnetic polarizabilities of the proton and studying the quark structure of nucleons, he now devotes his time and effort to the physics of baseball. He maintains an oft-visited website devoted to that subject: go.illinois.edu/physicsofbaseball.

When I woke up on Wednesday morning and checked my overnight Twitter timeline, I found considerable buzz about an incredible throw made by Oakland A’s left fielder Yoenis Cespedes on Tuesday night.

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With the game between the A’s and the Angels tied 1-1 and Howie Kendrick on first with one out in the bottom of the eighth, Mike Trout hit a fly ball to left field that fell in for a hit. When Cespedes misplayed the bounce and the ball rolled into foul territory, Kendrick kept on running. Cespedes eventually fielded the ball, over 300 feet from home plate, then uncorked a perfect strike on the fly to nab Kendrick at home plate. Prompted by several calls on the Twittersphere for a “physics analysis” of this amazing throw, I decided to forget about what I had been planning to do concentrate on this analysis instead. So, here we go.

What I wanted to do was come up with a way to determine the full trajectory of the throw. To do that, I need two critical pieces of information: How far did the ball travel (D), and how long was it in the air (T). To answer both questions, I replayed the video over and over again until I was satisfied that I had a good estimate of both D and T.

First, we know that the distance to the left-field foul pole in Anaheim is 340 ft. I estimate from the video that Cespedes was about 20 ft from the foul pole and just inside the LF line when he unloaded the ball. I also estimate that the ball was caught by A’s catcher Derek Norris about 2 ft from the corner of home plate. Thus I find that D=318 ft. Next, I used several different camera angles to time the throw with a stopwatch. I found amazing consistency with that process, arriving at T=3.17 sec. Now, I should point out that ESPN reported somewhat different numbers, D=300 ft and T=2.78 sec. I am pretty confident about the T, and perhaps less confident about the D, since I had to estimate the distance from the foul pole. But ESPN had to do that also. In any case, I’ll proceed with my numbers and comment later on doing the same analysis with the ESPN numbers.

The next step is to use my “trajectory calculator,” a general-purpose tool I have created to do all kinds of trajectory calculations for baseball and softball. While I normally use it for pitched or batted balls, there’s no reason why it can’t also be used for thrown balls, so that’s what I did.

First, I made certain assumptions about the drag coefficient and the amount of backspin on the ball. The specific assumptions are not terribly important, since the final result is not all that sensitive to reasonable changes in those assumptions. Based on the video, I assumed that the throw was released from a height of 6 ft and was caught at a height of 5 ft. I then simply adjusted the release speed and vertical launch angle to make the trajectory be 5 ft off the ground and 318 from release after 3.17 sec. I find that the ball was released at a speed of 97-99 mph and at a launch angle of 12-14 degrees. My best estimate of the trajectory is shown in the plot below. By the way, had I used the ESPN numbers for D and T, the initial speed and angle would be 99 mph and 10.5 degrees, which are not really all that different from my results. So, there seems to be no doubt that the release speed was in excess of 95 mph and the launch angle was quite low, in the 10-14 degree range.

Having satisfied myself that I had a good model for the full trajectory, I then asked another question: How accurate did Cespedes have to be to nab the runner at home plate? The replay shows that there was very little margin for error. How does that translate in the accuracy of release?

I used the trajectory calculator to ask what would be the effect of changing either the horizontal or vertical release angles by ±1 degree. I found that a ±1 degree change in horizontal angle would lead to a horizontal deflection of about ±6 ft at home plate, probably making it impossible to nail the runner. I found that at ±1 degree change in the vertical angle would change the height of the ball at home plate by ±5 ft, meaning the ball would have hit the ground just in front of the plate or nearly gone over the catcher’s head. So we can safely conclude that Cespedes’s margin of error was less than (but comparable to) ±1 degree in each direction.

To throw the ball that hard and that quickly (after all, he didn’t have time to “aim”) with that accuracy is truly an amazing feat. Everyone who has seen the throw knows that already, but now we’ve quantified exactly how amazing it was.

Thank you for reading

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Well, we've quantified how amazing it was in absolute terms, but not relatively. How can we know how hard it was for Cespedes to do that - either compared to other players or compared to his own prior performance? (Not that it wasn't completely amazing... I'm just curious how difficult it was, the opportunity for any OF to do this is extraordinarily rare, and in many cases - since it means airmailing the cut-off man - discouraged.)

I appreciate the article, and the analysis that u would be completely incapable of. And I was utterly as amazed by the throw as everyone else. But, reading the article, I kept wondering one thing. What were you going to do today before writing this article?

Really nicely done. This throw got me thinking: do any sites keep track of things like total number of throws to X base and what percentage were cut off/on target/off target? With Cespedes it's easy since he leads the bigs in outfield assists. But do we know who has the mos accurate arm in the OF from year to year, regardless of whether the throw made it in time or not?

This was a great throw from distance point of view but it pales in comparison to the Great Bo Jackson throw from the left field corner of Royals stadium to get a much superior runner in Harold Reynolds by an eye lash.
Cespedes kicked the ball around, missed the cut off man by a wide margin, and made a high arching through rather than a straight one.
I don't know who has the most accurate arm in the bigs but I would put my money on Alex Gordon.

Update: I have updated my estimates based on new information I have found (courtesy of Eli Baldrige of ESPN Sports Science). Following his advice, I have used the Distance Measurement Tool with (old) Google Maps to come up with a much better distance, D=300 ft. I had earlier found some incorrect info on the web about the distance to the foul plot, which is actually 330 ft. Also, I now have a real-time video that allows me to time the throw by counting frames, with 30 frames/sec. I find that the hang time T=2.80 sec. The longer time I had used earlier was found from a replay that had actually been slowed down.
So, with D-300, T=2.80, I find initial speed=101.5 mph, launch angle=10.0 degrees. The answer is not all that different from before, but I wanted to get the whole thing right. I have posted a new plot of the trajectory on my web site.

Tom Hart, sideline reporter on the Braves telecasts, quoted your information stat for stat on Thursday afternoon's Braves-Rockies broadcast attributing the information to Baseball Prospectus. His segment included replays of the throw. Pretty good given that the Braves weren't playing the A's and it wasn't related to anything going on with the Braves.

Outfielders throwing velocity is not truly compable to pitching off a mound because the motions are very different. In this case though, Cespedes was not able to bring his full momentum toward home plate as an outfielder camped under a flyball. Yes,this throw was amazing, but throwing 100+MPH from the outfield is nothing special from a MLB caliber OFer.

Re: "...throwing 100+MPH from the outfield is nothing special...". I don't necessarily doubt what you are saying. Just wondering how you know that. In particular, do you have access to data that show that this is the case?
My student, Eric Lang, is busy working on a similar analysis for other great throws (especially the ones pointed out by Rob Neyer last week in his article). If we are successful in obtaining good video, we should be able to do the same type of analysis we did on Cespedes. Then we can compare release speeds, etc.